I'm trying to understand what is the point of the local volatility model in practice. Rather than asking a question I will explain what is what for me hoping someone will spot where I'm wrong:
The point of using another model than Black-Scholes is "roughly" to capture more market informations in a better way obviously, in order to price derivatives mostly (even exclusively).
Now the main subject of "how to price derivatives" can be rephrase as "how to interpolate the implied volatility", am I right?
But since that, in order to calibrate the local volatility model, you need to interpolate the implied volatility first, how does using this model can make sence in practice?